”Faster than Light” Propagation of Laser Pulses • In a recent Nature article [1], researchers reported observing • superluminalpropagation of a laser pulse in a gain medium • by a new mechanism • The Nature article received a great deal of attention in both • the scientific community and the world press [1] L. J. Wang, A. Kuzmich, and A. Doggariu, Nature (London)406, 277 (2000)

signal velocity u Causality • Causality requires that the effect appears after the cause • According to special relativity, in a frame moving • with velocity v < c, the time interval between events is • Special relativity requires that the signal velocity be less than c • Causality is violated if signal velocity u is greater than c

Experimentally observed advancement of a laser pulse in a gain doublet[Wang, et al., Nature, 406, 277 (2000)] The Nature article claims that 1) “differs from previously studied anomalous dispersion associated with an absorption or a gain resonance” 2) “ the shape of the pulse is preserved” 3) “the argument that the probe pulse is advanced by amplification of its front edge does not apply”

Laser Pulse vg vph Phase and Group Velocity of Laser Pulses Phase Velocity is the velocity of the phase Group Velocity is the velocity of the pulse envelope • If the pulse envelope does not distort then the group velocity is the • velocity of energy flow

Two Level Atom, α > 0 Refractive Index • The refractive index represents the medium’s response to the fields • Using the Classical Lorentz or Quantum Two Level Atom model where α denotes population inversion, N is the density of atoms, Ω is the atomic binding frequency, Γ is the damping rate , q is the electronic charge and m the electronic mass

Phase velocity: where • Group velocity: Phase and Group Velocities • There are conflicting interpretations of how pulses propagate • when vg is “abnormal” , i.e., vg < 0 or vg > c

Probe pulse ׃ • Envelope of the probe pulse at resonance : Rabi frequency of coupling beam Probe Pulse has Extremely Low Group Velocity and Losses • The group velocity of the probe pulse is • No losses in probe pulse

Group Velocity vg dispersive medium z : refractive index Expand wavenumber about carrier frequency where is the group velocity

Assuming low damping • The real part of the refractive index near the • resonance frequency, • If the medium is amplifying ( < 0) then the • group velocity is superluminal ( vg > c), (“Optical Tachyons”) Superluminal Group Velocity

Laser Envelope Equation In frequency domain Using the representations for the fields Since is non zero for small positive values of

Standard Fourier Transform Approach In the standard Fourier Transform approach care must be taken in ordering the terms For example: Index is expanded to 1st order (GVD is neglected) Exponential factor contains contributions beyond the order of approximation

Envelope Equation Substituting E and P, and relationship between A(z,t) and B(z,t), into wave equation yields an envelope equation For narrow spectral pulse widths ( long pulses ) can keep terms to order ( lowest order in GVD ) Envelope Equation

Analysis Consistent Solution: Exponential is expanded to consistent order Pulse propagates at the speed of light and undergoes distortion. Front is preferentially amplified when k1 < 0, i.e., differential gain.

Analysis Integral solution: Standard Approach: Lowest order solution is obtained by neglecting k2 and carrying out the integral without expanding the exponential: [Refs.] Analysis results in a pulse propagating undistorted at the group velocity: